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Engineering and Physics University of Central Oklahoma Dr. Mohamed Bingabr

ENGR 4323/5323 Digital and Analog Communication. Ch2 Signals and Signal Space. Engineering and Physics University of Central Oklahoma Dr. Mohamed Bingabr. Outline. Size of a Signal Classification of Signals Useful Signals and Signal Operations Signals Versus Vectors

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Engineering and Physics University of Central Oklahoma Dr. Mohamed Bingabr

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  1. ENGR 4323/5323 Digital and Analog Communication Ch2 Signals and Signal Space Engineering and Physics University of Central Oklahoma Dr. Mohamed Bingabr

  2. Outline • Size of a Signal • Classification of Signals • Useful Signals and Signal Operations • Signals Versus Vectors • Correlation of Signals • Orthogonal Signal Sets • Trigonometric and Exponential Fourier Series

  3. Signal Energy and Power Energy Signal Power Signal Energy Power

  4. Signal Classification 1. Continuous-time and discrete-time signals 2. Analogue and digital signals 3. Periodic and aperiodic signals 4. Energy and power signals 5. Deterministic and probabilistic signals 6. Causal and non-causal 7. Even and Odd signals

  5. Analog continuous Digital continuous Analog Discrete Digital Discrete Periodic Deterministic Aperiodic Probabilistic

  6. Useful Signal Operation Time Shifting Time Scaling Time Inversion

  7. Useful Signals Unit impulse Signal Unit step function u(t)

  8. Signals Versus Vectors By sampling, a continuous signal g(t) can be represented as vector g. g = [ g(t1) g(t2)… g(tn)] Vector Approximation To approximate vector g using another vector x then we need to choose c that will minimize the error e. g = cx + e

  9. Signals Versus Vectors Value of c that minimizes the error Signal Approximation g(t) = cx(t) + e(t) x(t)

  10. Correlation of Signals Two vectors are similar if the angle between them is small. Correlation coefficient Note: Similarity between vectors or signals does not depend on the length of the vectors or the strength of the signals.

  11. Example 1=1 2=1 3=-1 4=0.961 5=0.628 6=0

  12. Correlation Functions Cross-correlation Function Autocorrelation Function

  13. Orthogonal Signal Sets Orthogonal Vector Space g = c1x1 + c2x2 + c3x3 Orthogonal Signal Space g(t) = c1x1(t)+ c2x2(t) + … + cNxN(t) Parseval’s Theorem

  14. Trigonometric Fourier Series

  15. Example f(t) 1 e-t/2 -p 0 p • Fundamental period T0 = p • Fundamental frequency f0 = 1/T0 = 1/p Hz w0 = 2p/T0 = 2 rad/s To what value does the FS converge at the point of discontinuity?

  16. Compact Trigonometric Fourier Series We can use the trigonometric identity a cos(x) + b sin(x) = c cos(x + ) to find the compact trigonometric Fourier series C0, Cn, and θnare related to the trigonometric coefficients an and bn as:

  17. Role of Amplitude in Shaping Waveform

  18. Role of the Phase in Shaping a Periodic Signal

  19. Compact Trigonometric f(t) 1 e-t/2 -p 0 p • Fundamental period T0 = p • Fundamental frequency f0 = 1/T0 = 1/p Hz w0 = 2p/T0 = 2 rad/s

  20. Line Spectra of x(t) • The amplitude spectrum of x(t) is defined as the plot of the magnitudes |Cn| versus  • The phase spectrum of x(t) is defined as the plot of the angles versus  • This results in line spectra • Bandwidth the difference between the highest and lowest frequencies of the spectral components of a signal.

  21. Line Spectra f(t) 1 e-t/2 -p 0 p f(t)=0.504 + 0.244 cos(2t-75.96o) + 0.125 cos(4t-82.87o) + 0.084 cos(6t-85.24o) + 0.063 cos(8t-86.24o) + … Cn n 0.504 0.244  0.125 0.084 0.063 -/2  0 2 4 6 8 10

  22. Dn = - D-n Exponential Fourier Series To find Dn multiply both side by and then integrate over a full period, m =1,2,…,n,… Dn is a complex quantity in general Dn=|Dn|ej D-n = Dn* |Dn|=|D-n| Odd Even D0 is called the constant or dc component of x(t)

  23. Dn = Cn Line Spectra in the Exponential Form • The line spectra for the exponential form has negative frequencies because of the mathematical nature of the complex exponent. |Dn| = 0.5 Cn

  24. Example f(t) 1 -p/2 -p -2p p/2 p 2p • Fundamental period T0 = 2p • Fundamental frequency f0 = 1/T0 = 1/2p Hz w0 = 2p/T0 = 1 rad/s Find the exponential Fourier Series for the square-pulse periodic signal.

  25. Dn Exponential Line Spectra |Dn| 1 1 1 1

  26. Example f(t) 1 -p/2 -p -2p p/2 p 2p The compact trigonometric Fourier Series coefficients for the square-pulse periodic signal.

  27. Relationships between the Coefficients of the Different Forms

  28. Example Find the exponential Fourier Series and sketch the corresponding spectra for the impulse train shown below. From this result sketch the trigonometric spectrum and write the trigonometric Fourier Series. Solution -2T0 -T0 T0 2T0

  29. x(t) 1 -p/2 -p -2p p/2 p 2p Rectangular Pulse Train Example Clearly x(t) satisfies the Dirichlet conditions. The compact trigonometric form is n odd

  30. Parseval’s Theorem • Let x(t) be a periodic signal with period T • The average powerP of the signal is defined as • Expressing the signal as it is also

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